Ethan Fromettl.masfak.ni.ac.rs/RADOVI MA14068/RISK OF SELECTING A... · Web viewFor the comparative analysis of this model, in the control sense, it is most convenient to apply the

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RISK OF SELECTING A TETRAHEDRON FOR FEM STRUCTURAL MODELING

Miomir JOVANOVIĆPredrag MILIĆ

Goran PETROVIĆ

Abstract This paper deals with the accuracy problem of applying the finite element of tetrahedron type The variable quality of modeling a solid continuum with a finite element ndash tetrahedron ndash is presented through several examples The analysis is directed towards the dependence of accuracy of basic finite elements on the character of external construction load The results of numerous comparative FEM and control experimental analyses are given The presented examples show where the precision of discrete models is compromised and warn against the possibility of presence of large errors in analyses

Key words Tetrahedron Brick accuracy FEM analyses error

1 INTRODUCTION

The modern tasks of construction analysis are today characterized by the highly complex geometry of models which is most often demanded by the design Nowadays the method of finite elements is in widespread practical use The constructions which are treated by this method are most frequently subjected to automatic procedures of generating finite element grids in order to reduce the model development time In automatic procedures of modern software the finite element tetrahedron is generally applied for discrete continuum modeling Such a small number of nodes compared to the brick (an eight-node element) has a varying success in describing the internal stress-strain state of constructions The reason for applying the tetrahedron is in the fluent description of the complex construction topology and the working speed of grid generator This familiar disadvantage of tetrahedron application is still present in modern models since designers find the automatic grid generation which is supported by the majority of software much more efficient On the other hand the demand for the model and analysis quality

imposes the need for constant comparative checking of model accuracy The experience from numerous analyses has shown that a good quality of modeling can be achieved using the tetrahedron which is most often in the function of tetrahedron position in relation to external action and the gradient of internal change of the stress-strain state Let us consider several simple comparative models in further text

2 MODELS SUBJECTED TO AXIAL FORCES

Pipe systems under pressure are characterized by high tensile stress and usually proportional expansion of continuum-walls (of slightly variable thickness) Let us observe a quarter of the pipe (with dimension DD0=25625 m and interior preassure p=567105 Pa) for feeding water to hydro plant turbines These pipe constructions are characterized by the proportional expansion of walls due to the internal water pressure The large diameters of pipelines and the equal loading usually cause membrane stress in walls and a very low degree of bending Starting from the geometry of concretely derived objects the example of determining the stress and movement (expansion) of pipes was made by applying the FEM analysis Four grids were developed in the example The first two (M1 and M2) were modeled with an eight-node finite element (brick) while the third and fourth (M3 and M4) were derived using a four-node and ten-node element (tetrahedron) At that the first third and fourth grid had the same element size which accounted for the equal number of grid nodes in M1 and M3 The M2 grid was developed for monitoring the effect of element size on model accuracy (convergence) There were 64 times more elements in grid M2 since 4 more elements were introduced into each of the element directions of grid M1 for the 3D model continuum space The analytical value of stress in the circumferential direction of the pipe was determined on the basis of Relation 1 [1] The theoretical stress on the inner pipe wall in the circumferential direction from the pressure on the inner pipe wall (thick walled tube) is

(1)

Comparing the accuracy of the use of the brick element (grid M1) and the use of the tetrahedron (grid M3) showed good agreement of the analyses the relative difference between the numerical and analytical results (1) was +0581 and +0964 at the same size of finite elements Table 1

Fig1 Stress in Circumferential Direction model M1According to this Table the stresses obtained by the tetrahedron were very close to the analytical results The finite element brick gave the results only slightly closer to the analytical solution than the stresses obtained by the tetrahedron in model M3 Model M4 gave even better results with more movement degrees of freedom and more nodes than model M1 and difference of +0385

Fig2 Stress in Circumferential Direction model M2

Figures 1 and 2 show the layout of the grid and circumferential stresses of models M1 and M2 derived by the eight-node element ndash brick

Figure 3 shows a detail of grid M3 generated by tetrahedrons It is obvious that in these constructive situations the free expansion of a pipe under internal pressure the tetrahedron represents the internal stress changes well The highest quality solution was obtained with the greatest number of movement degrees of freedom 252529 (model M2 with the finite element brick) and difference of +0170

XY Z

V1L1C1G1

Output Set MSCNASTRAN Case 1Deformed(000152) Total Translation

Fig3 Model M3 ndash tetrahedrons

Table 1

BRICK ELEMENT TETRA ELModel ID M1 M2 M3 M4Number of element 1040 66560 6240 6231Element size [mm] 30 75 30 30Node number 2244 84825 2244 12972Grid on model 1 x 65 x 16 4 x 260 x 64 1 x 65 x 16 1 x 65 x 16DOF 6532 252529 6532 38352Stress in Circumferential Direction C [KNcm2] 23911 23911 23911 23911

Numerical FEM solution C [KNcm2] stress

240499on element 28

239516on element 66673

241414 on element 7339


Relative difference numerical from analytical solution [] + 0581 + 0170 + 0964 + 0385

Ratio numerical and analytical stress solution [] 100581 100170 100964 100385

Total translation [m] 000151 000151 000152 000151

3 MODELS SUBJECTED TO BENDING

Let us observe a thick truck spring band subjected to a concentrated load in the middle and supported as a simple beam with a single moveable fulcrum The dimensions of the band are BxxL=008x001x08m and the arbitrary concentrated force F=200N Such an elastic form is characterized by a distinct presence of the bending moment As was the case in the previous example four analysis models were developed for this task M5 M6 M7 and M8 For the comparative analysis of this model in the control sense it is most convenient to apply the linear bending theory The component stress in the direction of band length determined in the middle of band span by the linear theory is defined by Relation 2

(2)Four grids were developed in the example The first two (M5 and M6) were modeled with an eight-node finite element (brick) while the third and fourth (M7 and M8) were derived using a four-node (tetrahedron) At that grids M5 and M7 had the same element size The M6 grid was developed for monitoring the effect of element size on model accuracy (convergence) There were 64 times more elements in grid M6 since 4 more elements were

introduced into each of the directions of the basic grid M5 for the 3D model continuum space Thereat the tiniest brick elements were generated by grid M6 The smallest approximations of the continuous model were made by that so that this model corresponded to the highest analysis accuracy This was done in compliance with the principle of small movements since the models were set in such a way that the translations (movements) of the nodes were smaller than the size of finite model elementsComparing the accuracy of the use of the brick element (grid M5) and the use of the tetrahedron (grid M7) showed a large difference in numerical values of FEM solutions The finite element brick in models M5 and M6 gave the stresses which differ for 1736 0743 from the analytical solutions The application of the tetrahedron in model M7 gave a difference of 71572 of the numerical in comparison to the analytical model The stresses of the numerical solution M7 (obtained by tetrahedrons) were significantly lower and they accounted for only 28428 of the analytical solution The differences between the solutions of numerical analyses (derived with the tetrahedron with four nodes) and the solutions of analytical analyses were so great that the former were considered inaccurate Grid M8 formed out of tetrahedrons with the middle node at the sides was of high quality since the number of nodes in the model was enlarged 6 times in relation to the starting model M5 The difference of model M5 (regardless of bending) was -0778 from the analytical solution This analysis of many grids shows that even in the case of bending a tetrahedron of ten nodes can be applied very successfully

Figure 4 shows the layout of the grid (Solid Von Mises stress) of model M5 derived by the eight-node element ndash brick Figure 5 shows a detail of grid M7 (Solid Von Mises stress) generated by tetrahedrons

X

Y

Z22222

123

2222222222222222222222222222222222222222

12121212 12121212 12121212 12121212 12121212 12121212 12121212 1212121212

30170288

28308958

26447628

24586298

22724969

20863639

19002309

17140979

15279649

13418319

11556989

9695660

7834330

5973000

4111670

2250340

389010

V1L1C1

Output Set MSCNASTRAN Case 1Deformed(00016) Total TranslationContour Solid Von Mises Stress

Fig4 Solid Von Mises stress M4 (Brick element)

X

Y

Z

9286880

8723701

8160523

7597344

7034165

6470987

5907808

5344629

4781451

4218272

3655093

3091915

2528736

1965557

1402379

839200

276021

V1L1C2


Fig 5 Solid Von Mises stress M6 (Tetra element)

Table 2

BRICK TETRAModel ID M5 M6 M7 M8Number of element 640 40960 3840 4219Element size [mm] 10 25 10 10Node number 1458 52965 1458 8716Grid on model 1 x 8 x 80 4 x 32 x 320 1 x 8 x 80 1 x 8 x 80DOF 4329 158730 4336 26063Analytic solution z [KNcm2]component stress 30000 30000 30000 30000

Numerical FEM solution z [KNcm2] component stress 294793 297772 085285 297666

Relative difference numerical from analytical solution [] - 1736 - 0743 - 71572 - 0778


4 PRACTICAL RESEARCH

Modeling done on real structures are the main reason for this paper In numerous professional analyses of researchers the difference between numerical results obtained by applying the FEM model with the tetrahedron and experimental results is determined An example related to pipe systems is shown in further text [2] A pre-turbine pipe system (fork) of a hydro-aggregate in a power plant was numerically analyzed The fact that the

modeled structure was of great mass and complex geometry with transitional wall thicknesses discontinuous pipe bends and seam details implied just the selection of a basic tetrahedron (with four nodes) as the element with which a really large discrete model could be generated (developed) Of course it is clear that generating a discrete model is in itself useless if it leads to incorrect or unconfirmed data Therefore the modeling requires the use of discrete models of proven quality or finding evidence of model accuracy [4 5] By this task a discrete model with the mass of 44333kg was automatically

generated by a usual processor [3] which worked efficiently only with four-node tetrahedrons The reduction of model error was searched for in the multi-layer grid and very small finite elements The FEM fork model grid was characterized by 1687170 elements whose average mass was 265 grams The size of the model and hardware limitations caused the application of four-node tetrahedrons for discrete modeling The application of a ten-node tetrahedron would have led to a model of tens of millions of finite elements which demanded a stronger hardware platform than a PC desktop working station and a superior grid generator

In order to check the stress-strain state a comparative experimental examination of the system in the stationary position under constant internal pressure was conducted Figure 6 shows the geometrical model of the fork in the pipe with the layout of measurement points for tensometric measurement

Table 3 shows an isolated comparison of interesting numerical (FEM) and experimental results on two characteristic places on the fork measurement point no 9 in the pipe seam zone (where the thickness of the wall changes) and measurement point no 10 on the ellipse (rib) of the fork The results of stresses agreement in applying the finite element tetrahedron (with four nodes) were good where dominant tension or pressure occurred Where bending occurred the obtained FEM stresses differed greatly from experimental ones A significant difference appeared in measurement point 9 as a consequence of the local bending of the transitional wall thickness thus making the application of tetrahedrons locally inadequate In measurement point 10 (the side of ellipse) dominant normal edge stresses (parallel to the contour) occurred which led to numerically (by tetrahedron) obtained results being experimentally confirmed

Table 3

Measurement point Element number

Von Mises numerical stress (FEM)[Nm2]

Equivalent experimental stress

(R1R2R3)[Nm2]

Relative difference numerical from experimental solution []

9 1688754 193464800 274333000 2947810 1688404 179536288 179497000 -0022

Fig 6 Fork and Pre-Turbine Pipe of Hydro-Aggregate [2]

Fig 5 Detail of model grid Fork Hydro-Aggregate

5 CONCLUSION

1 Tetrahedron as a finite element is most easily applicable in the discrete modeling software That is why it can be found in all grid modelers of finite element software This finite element must be used carefully as the example presented in this paper show

2 Tetrahedron does not show significant errors in structural analyses of a continuum subjected to axial loading These are pressure tanks and pipe systems without contact effects from other elements

3 The presence of bending in constructions leads to great differences in analyses accuracy which is usually evinced by significantly lower stresses than theoretical ones when applying four-node tetrahedrons Results obtained through ten-node tetrahedrons are of high quality

4 Differences in the quality of structural internal stresses modeling come from the manner in which the interpolation functions work within a tetrahedron The interpolation of the spatial stress state is conducted on the basis of data from the four nodes while it is conducted on the basis of eight nodes in a brick Also results obtained from ten-node tetrahedrons are much better than those from four-node models The degeneration of tetrahedrons in automatic grid generating can only deteriorate grid quality and lead to even greater differences

5 Finite elements method users must be well aware of the character of internal spreading of structural stress in order to be able to apply tetrahedron as modeling element

6 The ease that applicable software offers when applying tetrahedron as grid modeling element often leads to predicted the application possibilities of this finite element for the identification of the exact stress state

7 The versatility of tetrahedron positions in a complex continuum most frequently leads to a variable quality of model accuracy depending on the dominance of internal effects Where the transverse stresses are dominant in the continuum the analysis results are much better And where the bending stresses are dominant large errors are present just as this analysis shows

8 The application of tetrahedrons introduces the shortcomings of post-processing distribution of stress which can be observed by the dependence of stress on the topological form of a grid The distribution of stress certainly depends only on the geometrical form of the continuum and the loading manner not the grid topology

REFERENCES

[1] WATKINS R K ANDERSON L R Structural mechanics of buried pipes Utah State University Logan Utah CRC press LLC 2000

[2] Strain-stress investigation of characteristic cross section pipe system C3 in Power plant bdquoPerućicardquo-part 9 Mechanical Faculty Podgorica July 2008 (working material)

[3] MSC Nastran 2003 Linear static analysis MSCSoftware Corporation Santa Ana 2004

[4] BATHE K JWILSON E Numerical methods in finite element analysis Prentice Hall Englewood 1976

[5] ZIENKIEWICZ O C ZHU JZ Adaptivity and mesh generation International Journal for numerical methods in engineering science Vol32 783-810 1991

AcknowledgmentThis paper is financially supported by the Ministry of Science and Technological Development of Republic of Serbia Project Nr 14068 This support is gratefully acknowledged

CORRESPONDENCE

Miomir JOVANOVIĆ Prof DSc EngUniversity of NišFaculty of Mechnical EngineeringChair of Transport tec and LogisticsStr A Medvedeva 1418000 Niš Serbiamiomirmasfakniacrs

Predrag MILIĆ BSc EngUniversity of NišFaculty of Mechnical EngineeringChair of Transport tec and LogisticsStr A Medvedeva 1418000 Niš Serbiapmilicmasfakniacrs

Goran PETROVIĆ MSc EngUniversity of NišFaculty of Mechnical EngineeringChair of Transport tec and LogisticsStr A Medvedeva 1418000 Niš Serbiapgoranmasfakniacrs

Fig1 Stress in Circumferential Direction model M1According to this Table the stresses obtained by the tetrahedron were very close to the analytical results The finite element brick gave the results only slightly closer to the analytical solution than the stresses obtained by the tetrahedron in model M3 Model M4 gave even better results with more movement degrees of freedom and more nodes than model M1 and difference of +0385

Fig2 Stress in Circumferential Direction model M2

Figures 1 and 2 show the layout of the grid and circumferential stresses of models M1 and M2 derived by the eight-node element ndash brick

Figure 3 shows a detail of grid M3 generated by tetrahedrons It is obvious that in these constructive situations the free expansion of a pipe under internal pressure the tetrahedron represents the internal stress changes well The highest quality solution was obtained with the greatest number of movement degrees of freedom 252529 (model M2 with the finite element brick) and difference of +0170

XY Z

V1L1C1G1

Output Set MSCNASTRAN Case 1Deformed(000152) Total Translation

Fig3 Model M3 ndash tetrahedrons

Table 1

BRICK ELEMENT TETRA ELModel ID M1 M2 M3 M4Number of element 1040 66560 6240 6231Element size [mm] 30 75 30 30Node number 2244 84825 2244 12972Grid on model 1 x 65 x 16 4 x 260 x 64 1 x 65 x 16 1 x 65 x 16DOF 6532 252529 6532 38352Stress in Circumferential Direction C [KNcm2] 23911 23911 23911 23911

Numerical FEM solution C [KNcm2] stress

240499on element 28


241414 on element 7339


Relative difference numerical from analytical solution [] + 0581 + 0170 + 0964 + 0385


Total translation [m] 000151 000151 000152 000151

3 MODELS SUBJECTED TO BENDING

Let us observe a thick truck spring band subjected to a concentrated load in the middle and supported as a simple beam with a single moveable fulcrum The dimensions of the band are BxxL=008x001x08m and the arbitrary concentrated force F=200N Such an elastic form is characterized by a distinct presence of the bending moment As was the case in the previous example four analysis models were developed for this task M5 M6 M7 and M8 For the comparative analysis of this model in the control sense it is most convenient to apply the linear bending theory The component stress in the direction of band length determined in the middle of band span by the linear theory is defined by Relation 2

(2)Four grids were developed in the example The first two (M5 and M6) were modeled with an eight-node finite element (brick) while the third and fourth (M7 and M8) were derived using a four-node (tetrahedron) At that grids M5 and M7 had the same element size The M6 grid was developed for monitoring the effect of element size on model accuracy (convergence) There were 64 times more elements in grid M6 since 4 more elements were



X

Y

Z22222

123

2222222222222222222222222222222222222222

12121212 12121212 12121212 12121212 12121212 12121212 12121212 1212121212

30170288

28308958

26447628

24586298

22724969

20863639

19002309

17140979

15279649

13418319

11556989

9695660

7834330

5973000

4111670

2250340

389010

V1L1C1



X

Y

Z

9286880

8723701

8160523

7597344

7034165

6470987

5907808

5344629

4781451

4218272

3655093

3091915

2528736

1965557

1402379

839200

276021

V1L1C2



Table 2











Table 3




(R1R2R3)[Nm2]


9 1688754 193464800 274333000 2947810 1688404 179536288 179497000 -0022



5 CONCLUSION









REFERENCES







CORRESPONDENCE






X

Y

Z22222

123

2222222222222222222222222222222222222222

12121212 12121212 12121212 12121212 12121212 12121212 12121212 1212121212

30170288

28308958

26447628

24586298

22724969

20863639

19002309

17140979

15279649

13418319

11556989

9695660

7834330

5973000

4111670

2250340

389010

V1L1C1



X

Y

Z

9286880

8723701

8160523

7597344

7034165

6470987

5907808

5344629

4781451

4218272

3655093

3091915

2528736

1965557

1402379

839200

276021

V1L1C2



Table 2











Table 3




(R1R2R3)[Nm2]


9 1688754 193464800 274333000 2947810 1688404 179536288 179497000 -0022



5 CONCLUSION









REFERENCES







CORRESPONDENCE







Table 3




(R1R2R3)[Nm2]


9 1688754 193464800 274333000 2947810 1688404 179536288 179497000 -0022



5 CONCLUSION









REFERENCES







CORRESPONDENCE





5 CONCLUSION









REFERENCES







CORRESPONDENCE




REFERENCES







CORRESPONDENCE




Documents

Ethan Fromettl.masfak.ni.ac.rs/RADOVI MA14068/RISK OF SELECTING A... · Web viewFor the comparative analysis of this model, in the control sense, it is most convenient to apply the